Who is that?

He is my father, an engineer.

Who is this?
He is an engineer, my father.

Who is he?
He is my father.

Who is he?
He is an engineer.
--------------------------
Are all the answers correct?

Yes. They are all correct. But the second one bothers me because it puts his profession above his relationship to you.

Yes, all the answers are correct. In the first question, "He" refers to the person mentioned earlier, who is the speaker's father. In the second question, "He" also refers to the person mentioned earlier, who is an engineer and the speaker's father. In the third question, "He" once again refers to the speaker's father. In the fourth question, "He" refers to a different person who is an engineer.